Basically, was there an anti-American sentiment analogous to the Yellow Peril in Europe? Was there the idea of the American Peril? I know that America was not actually particularly militarily powerful at the time, but its industrial and economic influence would surely be felt, right?
I don't know if there was something strictly analogous to the concept of "Yellow Peril" but in the UK at the turn of the century there was a considerable fear of the US in terms of industrial and economic influence. I do not know whether this was a Europe-wide fear.
Britain by 1900 had dominated trade and industry for many decades but the lead over the other industrialising nations, especially Germany and the USA, had shrunk. The British public don't seem to have feared an American invasion - possibly due to distance - though they did fear a German one ("The Battle of Dorking" being a book that apparently epitomised "invasion scare literature", which was a genre back then), and they also seem to have been receptive to ideas about British trade and industry being swamped by America and Germany in particular.
A Liberal-turned-Conservative politician, Joseph Chamberlain, campaigned from 1903 for tariffs to be introduced on non-Empire goods; the logic was that this would protect British trade from foreign competition. As far as I know Chamberlain didn't specifically "gun" for one power (e.g., USA or Germany) being the main target but if you google "tariff reform posters" you'll find that several of them depict "American Workman" or "German Workman", or that boxes of cheap goods are marked "from USA" or "from Germany".
The problem for Chamberlain was that the British public essentially held free trade as an article of faith; one of the reason the Liberal Party did so well, and the Conservative Party didn't, in the 1906 General Election was that it was largely an election between a party that guaranteed free trade (Liberals) and a party that didn't necessarily guarantee free trade (Conservatives).